I wouldn't worry about it. Phones are consumables, like clothes. If you use them daily, eventually they get damaged or wear out. You can't fix the scratch. It probably cleans up with a cloth because you are cleaning the dirt/oils/etc that build up in it, but they will return.
I don't advise using a screen protector either as it degrades the experience. Screen does not feel as nice, can reduce sensitivity and image quality, and it looks like crap when it starts peeling up. Think about it, if a screen protector was really a good idea, why don't manufacturers install it from the factory with one? Costs probably a penny to make or less.
Keep on using the phone and have fun with it instead of worrying about a scratch. I would only worry about cosmetic stuff that affects the functionality of the phone (i.e. prevents from working properly) or looks REALLY bad (i.e. piece cracked off).